$\begingroup$$\int_0^a\cdots\,dx$ means that the value of $x$ goes from $0$ to $a$. If $x=a\tan\theta$ this means that the value of $\theta$ goes from $0$ to $\pi/4$. So you get $\int_0^{\pi/4}\cdots\,d\theta$.$\endgroup$
– DavidFeb 13 '18 at 5:11

$\begingroup$Once you square it, the terms inside the parentheses in the denominator become $a^2+a^2\tan^2\theta=a^2(1+\tan^2\theta)$, as shown by the next step.$\endgroup$
– Robert HowardFeb 13 '18 at 5:13